Different super-Eddington accretion models have been proposed in the literature so far
(see, e.g., the comprehensive review by [1]}).
One of the most widely adopted is the slim disc solution, a generalisation of the Shakura-Sunyaev thin disc solution, derived numerically integrating the Navier-Stokes equations [2]}, [3]}, that better describes super-critical regimes (\(\dot{M}>0.5 \dot{M_{\rm Edd}}\) ).
In the slim disc model, the super-Eddington flow of gas towards the BH proceeds via an optically thick accretion disc. The photons emitted by the accreting gas are trapped (i.e. can not escape the accretion flow) and are advected (dragged) inward, towards the central BH, by the accretion flow itself (see however [4]}), thus resulting in a radiative efficiency (\(\epsilon \) ) lower than the typical \(10\%\)  adopted in the standard optically thin disc model, \(0.002\lesssim \epsilon \lesssim 0.05\)  (independently of the BH spin), as shown by [5]}.
In other words, in this radiatively inefficient accretion regime, super-Eddington rates \(\dot{M}\le 100 \, \rm M_\odot /yr\)  translate in a only mildly super-Eddington disc luminosity \(L\sim (2-4) L_{\rm Edd}\)  and, consequently, the quantity \(1-\epsilon \)  is higher than in the Eddington-limited accretion scenario.
A lower radiative efficiency is also supported by observational evidences up to \(z>7\)  QSOs (e.g., [6]}, and references therein).
